This paper is devoted to the homogenization of a class of nonlinear nonlocal parabolic equations with time dependent coefficients in a periodic and stationary structure. In the first part, we consider the homogenization problem with a periodic structure. Inspired by the idea of Akagi and Oka for local nonlinear homogenization, by a change of unknown function, we transform the nonlinear nonlocal term in space into a linear nonlocal scaled diffusive term, while the corresponding linear time derivative term becomes a nonlinear one. By constructing some corrector functions, for different time scales $ r $ and the nonlinear parameter $ p $, we obtain that the limit equation is a local nonlinear diffusion equation with coefficients depending on $ r $ and $ p $. In addition, we also consider the homogenization of the nonlocal porous medium equation with non negative initial values and get similar homogenization results. In the second part, we consider the previous problem in a stationary environment and get some similar homogenization results. The novelty of this paper is two folds. First, for the determination equation with a periodic structure, our study complements the results in literature for $ r = 2 $ and $ p = 1 $. Second, we consider the corresponding equation with a stationary structure.
Citation: Junlong Chen, Yanbin Tang. Homogenization of nonlinear nonlocal diffusion equation with periodic and stationary structure[J]. Networks and Heterogeneous Media, 2023, 18(3): 1118-1177. doi: 10.3934/nhm.2023049
This paper is devoted to the homogenization of a class of nonlinear nonlocal parabolic equations with time dependent coefficients in a periodic and stationary structure. In the first part, we consider the homogenization problem with a periodic structure. Inspired by the idea of Akagi and Oka for local nonlinear homogenization, by a change of unknown function, we transform the nonlinear nonlocal term in space into a linear nonlocal scaled diffusive term, while the corresponding linear time derivative term becomes a nonlinear one. By constructing some corrector functions, for different time scales $ r $ and the nonlinear parameter $ p $, we obtain that the limit equation is a local nonlinear diffusion equation with coefficients depending on $ r $ and $ p $. In addition, we also consider the homogenization of the nonlocal porous medium equation with non negative initial values and get similar homogenization results. In the second part, we consider the previous problem in a stationary environment and get some similar homogenization results. The novelty of this paper is two folds. First, for the determination equation with a periodic structure, our study complements the results in literature for $ r = 2 $ and $ p = 1 $. Second, we consider the corresponding equation with a stationary structure.
| [1] | D. Cioranescu, P. Donato, An Introduction to Homogenization, Oxford: Oxford University Press, 2000. |
| [2] |
G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608–623. https://doi.org/10.1137/0520043 doi: 10.1137/0520043
|
| [3] |
G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482–1518. https://doi.org/10.1137/0523084 doi: 10.1137/0523084
|
| [4] |
A. Holmbom, Homogenization of parabolic equations an alternative approach and some corrector-type results, Appl. Math., 42 (1997), 321–343. https://doi.org/10.1023/A:1023049608047 doi: 10.1023/A:1023049608047
|
| [5] |
G. Akagi, T. Oka, Space-time homogenization for nonlinear diffusion, J. Differ. Equ., 358 (2023), 386–456. https://doi.org/10.1016/j.jde.2023.01.044 doi: 10.1016/j.jde.2023.01.044
|
| [6] | G. Akagi, T. Oka, Space-time homogenization problems for porous medium equations with nonnegative initial data, arXiv preprint, 2021. https://doi.org/10.48550/arXiv.2111.05609 |
| [7] |
J. Geng, Z. Shen, Convergence rates in parabolic homogenization with time-dependent periodic coefficients, J. Funct. Anal., 272 (2017), 2092–2113. https://doi.org/10.1016/j.jfa.2016.10.005 doi: 10.1016/j.jfa.2016.10.005
|
| [8] |
W. Niu, Y. Xu, A refined convergence result in homogenization of second order parabolic systems, J. Differ. Equ., 266 (2019), 8294–8319. https://doi.org/10.1016/j.jde.2018.12.033 doi: 10.1016/j.jde.2018.12.033
|
| [9] | V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Averaging of parabolic operators, Trudy Moskov. Mat. Obshch., 45 (1982), 182–236. |
| [10] |
S. M. Kozlov, The averaging of random operators, Math. Sb., 109 (1979), 188–202. https://doi.org/10.1070/SM1980v037n02ABEH001948 doi: 10.1070/SM1980v037n02ABEH001948
|
| [11] |
M. Kleptsyna, A. Piatnitski, A. Popier, Homogenization of random parabolic operators, Stoch. Process. Their Appl., 125 (2015), 1926–1944. https://doi.org/10.1016/j.spa.2014.12.002 doi: 10.1016/j.spa.2014.12.002
|
| [12] | M. Kleptsyna, A. Piatnitski, A. Popier, Asymptotic decomposition of solutions to random parabolic operators with oscillating coefficients, arXiv, 2020. https://doi.org/10.48550/arXiv.2010.00240 |
| [13] |
A. Piatnitski, E. Zhizhina, Periodic homogenization of nonlocal operators with a convolution type kernel, SIAM J. Math. Anal., 49 (2017), 64–81. https://doi.org/10.1137/16M1072292 doi: 10.1137/16M1072292
|
| [14] |
A. Piatnitski, E. Zhizhina, Homogenization of biased convolution type operators, Asymptot. Anal., 115 (2019), 241–262. https://doi.org/10.3233/ASY-191533 doi: 10.3233/ASY-191533
|
| [15] | M. Kassmann, A. Piatnitski, E. Zhizhina, Homogenization of Levy-type operators with oscillating coefficients, SIAM J. Math. Anal., 51 (2019) 3641–3665. https://doi.org/10.1137/18M1200038 |
| [16] |
G. Karch, M. Kassmann, M. Krupski, A framework for nonlocal, nonlinear initial value problems, SIAM J. Math. Anal., 52 (2020), 2383–2410. https://doi.org/10.1137/19M124143X doi: 10.1137/19M124143X
|
| [17] | C. Cortazar, M. Elgueta, S. Martinez, J. D. Rossi, Random walks and the porous medium equation, Rev. Un. Mat. Argentina, 50 (2009), 149–155. |
| [18] |
F. Andreu, J. M. Mazon, J. D. Rossi, J. Toledo, The Neumann problem for nonlocal nonlinear diffusion equations, J. Evol. Equ., 8 (2008), 189–215. https://doi.org/10.1007/s00028-007-0377-9 doi: 10.1007/s00028-007-0377-9
|
| [19] |
A. de Pablo, F. Quirós, A. Rodríguez, J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378–1409. https://doi.org/10.1016/j.aim.2010.07.017 doi: 10.1016/j.aim.2010.07.017
|
| [20] |
A. de Pablo, F. Quirós, A. Rodríguez, J. L. Vázquez, A general fractional porous medium equation, Commun. Pure Appl. Math., 65 (2012), 1242–1284. https://doi.org/10.1002/cpa.21408 doi: 10.1002/cpa.21408
|
| [21] |
M. Bonforte, A. Figalli, X. Ros-Oton, Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains, Commun. Pure Appl. Math., 70 (2017), 1472–1508. https://doi.org/10.1002/cpa.21673 doi: 10.1002/cpa.21673
|
| [22] |
X. Yang, Y. Tang, Decay estimates of nonlocal diffusion equations in some particle systems, J. Math. Phys., 60 (2019), 043302. https://doi.org/10.1063/1.5085894 doi: 10.1063/1.5085894
|
| [23] |
C. Gu, Y. Tang, Chaotic characterization of one dimensional stochastic fractional heat equation, Chaos Solitons Fractals, 145 (2021), 110780. https://doi.org/10.1016/j.chaos.2021.110780 doi: 10.1016/j.chaos.2021.110780
|
| [24] |
C. Gu, Y. Tang, Global solution to the Cauchy problem of fractional drift diffusion system with power-law nonlinearity, Netw. Heterog. Media, 18 (2023), 109–139. http://dx.doi.org/10.3934/nhm.2023005 doi: 10.3934/nhm.2023005
|
| [25] |
M. Bonforte, J. Endal, Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities, J. Funct. Anal., 284 (2023), 109831. https://doi.org/10.1016/j.jfa.2022.109831 doi: 10.1016/j.jfa.2022.109831
|
| [26] | M. Bonforte, P. Ibarrondo, M. Ispizua, The Cauchy-Dirichlet problem for singular nonlocal diffusions on bounded domains, arXiv, 2022. https://doi.org/10.48550/arXiv.2203.12545 |
| [27] |
G. Beltritti, J. D. Rossi, Nonlinear evolution equations that are non-local in space and time, J. Math. Anal. Appl., 455 (2017), 1470–1504. https://doi.org/10.1016/j.jmaa.2017.06.059 doi: 10.1016/j.jmaa.2017.06.059
|
| [28] |
I. Kim, K. H. Kim, P. Kim, An $L^p-$theory for diffusion equations related to stochastic processes with non-stationary independent increment, Trans. Am. Math. Soc., 371 (2019), 3417–3450. https://doi.org/10.1090/tran/7410 doi: 10.1090/tran/7410
|
| [29] | F. Andreu, J. M. Mazon, J. D. Rossi, J. Toledo, Nonlocal Diffusion Problems, Providence: American Mathematical Society, 165 (2010), 256. |
| [30] | E. Zeidler, Nonlinear Functional Analysis and Its Applications: II/B: Nonlinear Monotone Operators, New York: Springer, 2013. https://doi.org/10.1007/978-1-4612-0981-2 |
| [31] | P. Benilan, M. G. Crandall, M. Pierre, Solutions of the porous medium equation in $R^{N}$ under optimal conditions on initial values, Indiana Univ. Math. J., 33 (1984), 51–87. http://www.jstor.org/stable/45010755 |
| [32] | P. Daskalopoulos, C. E. Kenig, Degenerate Diffusions: Initial Value Problems and Local Regularity Theory, Zurich: European Mathematical Society, 2007. https://doi.org/10.4171/033 |
| [33] | G. Leoni, A First Course in Sobolev Spaces, Providence: American Mathematical Society, 105 (2017), 607. |
| [34] |
J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1986), 65–96. https://doi.org/10.1007/BF01762360 doi: 10.1007/BF01762360
|
| [35] |
Z. Peng, Existence and regularity results for doubly nonlinear inclusions with nonmonotone perturbation, Nonlinear Anal. Theory Methods Appl., 115 (2015), 71–88. https://doi.org/10.1016/j.na.2014.12.010 doi: 10.1016/j.na.2014.12.010
|
| [36] |
de Pablo A, Quirs F, Rodrguez A, et al., A general fractional porous medium equation, Commun. Pure Appl. Math., 65 (2012), 1242–1284. https://doi.org/10.1002/cpa.21408 doi: 10.1002/cpa.21408
|
| [37] |
J. L. Vzquez, B. Volzone, Optimal estimates for fractional fast diffusion equations, J. Math. Pures Appl., 103 (2015), 535–556. https://doi.org/10.1016/j.matpur.2014.07.002 doi: 10.1016/j.matpur.2014.07.002
|
| [38] | J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs, Oxford: Oxford University Press, 2007. https://doi.org/10.1093/acprof: oso/9780198569039.001.0001 |
| [39] | J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type, Oxford: Oxford University Press, 2006. https://doi.org/10.1093/acprof: oso/9780199202973.001.0001 |
| [40] | M. G. Krein, M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk, 3 (1948), 3–95. https://www.mathnet.ru/eng/rm8681 |
| [41] |
Piatnitski A, Zhizhina E, Stochastic homogenization of convolution type operators, J. Math. Pures Appl., 134 (2020), 36–71. https://doi.org/10.1016/j.matpur.2019.12.001 doi: 10.1016/j.matpur.2019.12.001
|
| [42] |
P. Cardaliaguet, N. Dirr, P. E. Souganidis, Scaling limits and stochastic homogenization for some nonlinear parabolic equations, J. Differ. Equ., 307 (2022), 389–443. https://doi.org/10.1016/j.jde.2021.10.057 doi: 10.1016/j.jde.2021.10.057
|
| [43] |
E. Kosygina, S. R. S. Varadhan, Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium, Commun. Pure Appl. Math., 61 (2008), 816–847. https://doi.org/10.1002/cpa.20220 doi: 10.1002/cpa.20220
|
Junlong Chen, Yanbin Tang. Homogenization of nonlinear nonlocal diffusion equation with periodic and stationary structure[J]. Networks and Heterogeneous Media, 2023, 18(3): 1118-1177. doi: 10.3934/nhm.2023049